a ferris wheel has a radius of 10m and is one meter above the ground. If the ferris wheel makes 1 revolution every 20 seconds, write an equation that gives the height above the ground of a person on the ferris wheel as a function of time if that person starts (t=0) 1/8th of a rotation away from the bottom most point while moving downwards
please help!!
1 rotation = 2pi
1/8 rotation = 1/8 x 2pi = pi/4
State y as the height difference created from the rotation of the wheel and the bottom of the ferris wheel (not the ground)
height = y + 1
= r - r cos {(2pi/20)t - (pi/4)} +1
= r(1 - cos pi{(t/10) - (1/4)}) +1
= 10(1 - cos pi{(t/10) - (1/4)}) +1
To find an equation that gives the height above the ground of a person on the ferris wheel as a function of time, we can start by breaking down the problem into different components.
1. Determine the period of the ferris wheel:
The period of the ferris wheel is the time it takes to complete one full rotation. In this case, the ferris wheel takes 20 seconds to make one revolution.
2. Determine the height of the person at the bottommost point:
At the bottommost point, the person is one meter above the ground.
3. Determine the vertical displacement or amplitude:
The vertical displacement of the person on the ferris wheel is the difference in height between the highest and lowest points on the wheel. In this case, the amplitude is 10 meters - 1 meter = 9 meters.
4. Determine the phase shift:
The phase shift represents the amount of time it takes for the person to reach the bottommost point after starting. In this case, the person starts 1/8th of a rotation away from the bottommost point while moving downwards. Since 1 full rotation is equal to 2π radians, 1/8th of a rotation is equal to (1/8) * 2π = π/4. Therefore, the phase shift is π/4 radians.
5. Use the equation of a sinusoidal function:
The equation that represents the height of the person above the ground is given by:
h(t) = A * sin(B * (t - C)) + D
where:
- A is the amplitude (9 meters)
- B is the angular frequency (2π divided by the period, 2π/20)
- C is the phase shift (π/4 radians)
- D is the vertical shift (height at the bottommost point, 1 meter)
Combining all the values, we can write the equation as:
h(t) = 9 * sin((2π/20) * (t - π/4)) + 1
This equation gives the height above the ground of a person on the ferris wheel as a function of time.
To find the equation that gives the height above the ground of a person on the ferris wheel as a function of time, we need to consider the circular motion of the ferris wheel along with the starting position and direction of the person.
Let's first understand the motion of the ferris wheel. It makes one revolution every 20 seconds, which means it completes a full circle in 20 seconds. The radius of the ferris wheel is 10 meters, which means the diameter (2 times the radius) is 20 meters.
Now, since the person starts 1/8th of a rotation away from the bottommost point while moving downwards, that means the person is at a height of 10 meters (1 radius) above the ground. This is because the person is starting on the downward side of the ferris wheel when it is at its lowest point.
To write the equation for the height above the ground as a function of time, we need to consider the circular motion as well as the starting position and direction of the person.
The equation for the height (h) above the ground at any given time (t) can be written as follows:
h(t) = R * cos(2π * (t - t0) / T) + h0
Where:
- R is the radius of the ferris wheel (10 meters)
- t is the time in seconds
- t0 is the time offset (starting time) in seconds
- T is the period of the ferris wheel (20 seconds)
- h0 is the initial height of the person above the ground (10 meters)
In this case, since the person starts at 1/8th of a rotation away from the bottommost point while moving downwards, the time offset (t0) would be 1/8th of the period (T/8), which is 20/8 = 2.5 seconds.
Substituting the given values, the equation becomes:
h(t) = 10 * cos(2π * (t - 2.5) / 20) + 10
Now, you can plug in any value of time (t) into this equation to calculate the height above the ground at that particular time.